The simultaneous asymmetric perturbation method for overdetermined free boundary problems

TL;DR

This paper presents a novel perturbation method for overdetermined free boundary problems, enabling the construction of solutions with improved regularity and revealing geometric properties like symmetry and convexity.

Contribution

Introduces a simultaneous asymmetric perturbation approach that enhances solution regularity and expands the class of solvable overdetermined free boundary problems.

Findings

Constructed solutions with higher regularity.

Established geometric properties such as symmetry and convexity.

Improved understanding of free boundary regularity gaps.

Abstract

In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions that would have been out of reach otherwise. Another benefit of this method lies in the improvement of the regularity gap between the free boundary and the given one. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.